qReduMIS: A Quantum-Informed Reduction Algorithm for the Maximum Independent Set Problem

TL;DR

MIS and MWIS on graphs are NP-hard, motivating hybrid strategies that scale to large instances. qReduMIS fuses exact classical kernelization with a quantum co-processor to identify frozen vertices and unblock reductions, within a hardware-agnostic three-layer framework (application, algorithm, hardware). It defines a three-module pipeline—$\textsc{ClassicalReduce}$, $\textsc{QuantumMIS}$, and $\textsc{Select}$—that iteratively shrinks the problem, samples low-energy MIS candidates, and uses measurement statistics to guide further reductions, ultimately reducing to a kernel that can be solved exactly. Experiments on up to $n=231$ with a quantum backend based on Rydberg atom arrays show that qReduMIS can outperform purely quantum approaches, achieving $P_{\mathrm{MIS}}=1$ on hard cases and substantial gains on larger benchmarks, illustrating a practical quantum-classical synergy for MIS/MWIS with cross-platform potential.

Abstract

We propose and implement a quantum-informed reduction algorithm for the maximum independent set problem that integrates classical kernelization techniques with information extracted from quantum devices. Our larger framework consists of dedicated application, algorithm, and hardware layers, and easily generalizes to the maximum weight independent set problem. In this hybrid quantum-classical framework, which we call qReduMIS, the quantum computer is used as a co-processor to inform classical reduction logic about frozen vertices that are likely (or unlikely) to be in large independent sets, thereby opening up the reduction space after removal of targeted subgraphs. We systematically assess the performance of qReduMIS based on experiments with up to 231 qubits run on Rydberg quantum hardware available through Amazon Braket. Our experiments show that qReduMIS can help address fundamental performance limitations faced by a broad set of (quantum) solvers including Rydberg quantum devices. We outline implementations of qReduMIS with alternative platforms, such as superconducting qubits or trapped ions, and we discuss potential future extensions.

Figures (6)

• Figure 1: Schematic illustration of the hybrid qReduMIS framework, with distinct (i) application, (ii) algorithm, and (iii) hardware layers (from top to bottom). The application layer is illustrated for an example portfolio selection problem (PSP) that can be framed as a MIS (or MWIS) problem on an asset graph, the solution to which can be fed into a downstream portfolio composition problem. For a given problem instance on a graph $\mathcal{G}$, the qReduMIS algorithm solves the MIS problem by intermixing exact, polynomial time reduction logic with information obtained from quantum measurements to unblock the kernelization process whenever necessary by identifying and removing frozen vertices that have a high (or low) likelihood to be part of large independent sets. The qReduMIS framework is hardware-agnostic, and can be powered by both digital or analog quantum devices, such as Rydberg atom arrays implementing analog Hamiltonian simulation (AHS) programs.
• Figure 2: Snapshots of the qReduMIS algorithm (with in-set selection strategy) for the hardest instance listed in Tab. \ref{['tab:4hp']} with $\mathbb{H} \sim 1435$. (a) The problem input is given in terms of a site-diluted union-jack graph with $n=137$ nodes on a square lattice of side length $L=14$. (b) qReduMIS first calls $\textsc{ClassicalReduce}$. Orange nodes are selected ($n_{i}=1$), blue nodes are removed ($n_{i}=0$) per kernelization, until an irreducible kernel $\mathcal{K}$ is found, with $37$ kernel nodes displayed in dark gray. (c) Next, the QPU is called to unblock the reduction procedure. Solution candidates for the kernel are sampled, and the node highlighted in pink is identified as a frozen node with high in-set probability. Selection of this node and removal of its kernel neighborhood (pink box) results in an updated kernel $\mathcal{K}'$ that features an exposed corner node (highlighted with an orange dot). (d) The remaining kernel is fully reducible. Within two iterations (with two calls to $\textsc{ClassicalReduce}$ and one quantum-informed selection) qReduMIS finds an optimal solution of size $|\mathrm{MIS}|=45$.
• Figure 3: Success probability $P_{\mathrm{MIS}}$ as a function of the hardness parameter $\mathbb{H}$, for QAA (red squares), qReduMIS (blue circles), and a random-informed baseline (black triangles). Error bars refer to $90\%$ confidence intervals as extracted via bootstrapping.
• Figure 4: Performance as measured in terms of the independent set size $|\mathrm{IS}|$ for both QAA and qReduMIS (with in-set selection) for the two hardest instances considered in Tab. \ref{['tab:4hp']} with $\mathbb{H} \sim 125.5$ (left) and $\mathbb{H} \sim 1435$ (right), respectively. The optimal MIS sizes are given in the rightmost bin. (Left) QAA finds the optimum $|\mathrm{MIS}|=46$ with a small success rate of $P_{\mathrm{MIS}} \sim 0.7\%$, displaying a broad shoulder towards suboptimal solutions, while qReduMIS finds the optimum with a success rate of $P_{\mathrm{MIS}}=100\%$. (Right) The best solution found by QAA is $|\mathrm{IS}|=44$, while qReduMIS finds the MIS of size $|\mathrm{MIS}|=45$ with a success rate of $P_{\mathrm{MIS}} = 100\%$. Error bars refer to $90\%$ confidence intervals as extracted via bootstrapping, and $n_{\mathrm{shots}}=1000$ for all QPU calls.
• Figure 5: Number of QPU calls for both in-set and out-set selection strategy (with $\lambda=1$) for the two hardest instances considered in Tab. \ref{['tab:4hp']} with $\mathbb{H} \sim 125.5$ (top) and $\mathbb{H} \sim 1435$ (bottom), respectively.